You've decided to paint red and yellow flames on the side of your motorcycle. You want to blend the colors along each flare according to a mathematical formula involving hyperbolic trigonometric functions. Anything less than perfection would render the whole paint job useless. What's the best approach?
Surface Coatings for Fighter Aircraft
Leading aerospace engineers faced a similar problem in applying a surface coating to fighter aircraft. The critical feature in this case is not color but the electrical conductivity at the surface. This physical property determines how an electromagnetic wave will scatter when it hits the aircraft. But different parts of the aircraft have different levels of surface conductivity, and there's the catch--if there are sharp changes in conductivity from one area to another, the incoming wave scatters in a way enemy radar can detect, thus giving away the fighter's position.
To avoid this problem, airframers would spray a conductive surface coating of varying thickness onto the aircraft's surface, blending the edges so that no sudden transitions occur in surface resistance. Although the mathematical properties of the ideal blending pattern are known--they're a direct consequence of the laws of electrodynamics--applying the conductive coating properly still required a technician with a spray gun to create gradually blending areas exactly the way an airbrush artist paints flames on a motorcycle.
At least, that's how they used to do it, until they added Mathematica to their process. Mathematica is no stranger to either higher math or the creation of sophisticated graphic images.
Mathematica: A More Accurate Way to Apply Coatings
The process now goes like this. Engineers use Mathematica's numerical power to precisely define the ideal coating pattern for a given surface. Next, using a very high resolution PostScript printer, technicians print out a "phototool," an optical mask, from Mathematica. The phototool is then used in a photochemical etching process to place the conductive coating exactly where it's needed and with exactly the right thickness.
For some fighter parts, the ideal coating varies along the surface according to the hyperbolic cosine function. Because Mathematica is as familiar with hyperbolic trigonometric functions as it is with literally hundreds of other special mathematical functions, it could handle the task easily. The tight integration among Mathematica's algebraic, numeric, and graphics capabilities makes the path from initial mathematical model to final phototool a single, smooth development process.